P-form electrodynamics

In theoretical physics, p-form electrodynamics is a generalization of Maxwell's theory of electromagnetism.

We have a 1-form ${\displaystyle \mathbf {A} }$, a gauge symmetry

where ${\displaystyle \alpha }$ is any arbitrary fixed 0-form and ${\displaystyle d}$ is the exterior derivative, and a gauge-invariant vector current ${\displaystyle \mathbf {J} }$ with density 1 satisfying the continuity equation

where ${\displaystyle {\star }}$ is the Hodge star operator.

Alternatively, we may express ${\displaystyle \mathbf {J} }$ as a closed (n − 1)-form, but we do not consider that case here.

${\displaystyle \mathbf {F} }$ is a gauge-invariant 2-form defined as the exterior derivative ${\displaystyle \mathbf {F} =d\mathbf {A} }$.

${\displaystyle \mathbf {F} }$ satisfies the equation of motion

(this equation obviously implies the continuity equation).